Problem: Factor the following expression: $9$ $x^2$ $-41$ $x$ $-20$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(-20)} &=& -180 \\ {a} + {b} &=& & & {-41} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-180$ and add them together. Remember, since $-180$ is negative, one of the factors must be negative. The factors that add up to ${-41}$ will be your ${a}$ and ${b}$ When ${a}$ is ${4}$ and ${b}$ is ${-45}$ $ \begin{eqnarray} {ab} &=& ({4})({-45}) &=& -180 \\ {a} + {b} &=& {4} + {-45} &=& -41 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {9}x^2 +{4}x {-45}x {-20} $ Group the terms so that there is a common factor in each group: $ ({9}x^2 +{4}x) + ({-45}x {-20}) $ Factor out the common factors: $ x(9x + 4) - 5(9x + 4) $ Notice how $(9x + 4)$ has become a common factor. Factor this out to find the answer. $(9x + 4)(x - 5)$